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Showing papers on "Path graph published in 1976"


Journal ArticleDOI
TL;DR: In this paper, it was shown that if G is a planar graph on N vertices, then α(G)/N ⩾ 29 is the maximum number of vertices in an independent set.

28 citations


Journal ArticleDOI
TL;DR: A graph is strongly path connected if between each pair of distinct vertices there exist paths of all lengths greater than or equal to the distance between the vertices as discussed by the authors, i.e., there exists a path of length greater than and equal to

22 citations


Journal ArticleDOI
TL;DR: In this article, Harary et al. determined the least number of edges in a graphG in order to insure that G has propertyp(r, s) ifG contains a path of lengthr and if every such path is contained in a circuit of lengths.
Abstract: A graphG withn vertices has propertyp(r, s) ifG contains a path of lengthr and if every such path is contained in a circuit of lengths. G. A. Dirac and C. Thomassen [Math. Ann.203 (1973), 65–75] determined graphs with propertyp(r,r+1). We determine the least number of edges in a graphG in order to insure thatG has propertyp(r,s), we determine the least number of edges possible in a connected graph with propertyp(r,s) forr=1 and alls, forr=k ands=k+2 whenk=2, 3, 4, and we give bounds in other cases. Some resulting extremal graphs are determined. We also consider a generalization of propertyp(2,s) in which it is required that each pair of edges is contained in a circuit of lengths. Some cases of this last property have been treated previously by U. S. R. Murty [inProof Techniques in Graph Theory, ed. F. Harary, Academic Press, New York, 1969, pp. 111–118].

10 citations



Journal ArticleDOI
TL;DR: It is shown that a graph with no multiple edges on n vertices, n >= 5, with 2(n-2) arcs labelled 1,..., n-1 and 1',...,n-1' having at least one spanning tree whose arcs include no pair (j,[email protected]?), has at least six of them.

2 citations